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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 253704.bm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 253704.bm1 | 253704bm4 | \([0, 1, 0, -453912, -117843408]\) | \(5690357426/891\) | \(1619488316971008\) | \([2]\) | \(1966080\) | \(1.9294\) | |
| 253704.bm2 | 253704bm2 | \([0, 1, 0, -31072, -1477840]\) | \(3650692/1089\) | \(989687304815616\) | \([2, 2]\) | \(983040\) | \(1.5828\) | |
| 253704.bm3 | 253704bm1 | \([0, 1, 0, -11852, 474912]\) | \(810448/33\) | \(7497631097088\) | \([2]\) | \(491520\) | \(1.2363\) | \(\Gamma_0(N)\)-optimal |
| 253704.bm4 | 253704bm3 | \([0, 1, 0, 84248, -9780880]\) | \(36382894/43923\) | \(-79834775921793024\) | \([2]\) | \(1966080\) | \(1.9294\) |
Rank
sage: E.rank()
The elliptic curves in class 253704.bm have rank \(1\).
Complex multiplication
The elliptic curves in class 253704.bm do not have complex multiplication.Modular form 253704.2.a.bm
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
